Question 163
Let $ABC$ be a triangle for which $\angle BAC = 60^\circ$. Let $P$ ($Q$) be the point of intersection of the bisector of $\angle ABC$ ($\angle ACB$) and the side $AC$ ($AB$), respectively. Denote by $r_1$ ($r_2$) the radius of the in-circle of the triangle $ABC$ ($APQ$), respectively. Determine the radius of the circum-circle of the triangle $APQ$ in terms of $r_1$ and $r_2$.